2 Solving linear systems

we're interested iu solving the linear system

Ax b AE IR BE IN MEN

I itRecall's A e IR

h AA A A Iinverse exists if detCA 0

we call these matrices nonsing a Fae or

Cmatrix ainvertible

Gamel'srule det Aib with i th columnreplacedby b

diet A

requires htt determinants which isexpensive computationally

Computing the determinant requiresn operations

ie additions summationsfloatingpoint operations flops

2 Gaussian elimination

example III H 1 EEGenerate triangularsystem by adding multiples ofrows to other rows this does not change the solution

addC2 firstnow

us III of xto3rdrow

this is identical to multiplying the systemwith

4 12 9 and Lz I 9 fromtheleft

Ita E

c sixsecond2 E go

I us E

IsamuLig g's fly

IfTuppertriangularmatrix

A can be solved by backwards substitution

4 La downtriangular x 4matrices

Defi LE IR lowutriangulai if Lig O for allios

unitlowutriangulae if additionallyhi tha

Analogue for upper triangular unit uppa triangularmatrices

Thmi properties of lower triangular matrices identical resultsholdfor upper triangular

i products of lower triangularmollies

matrices are lowee triangularii as above for unit lower triangulariii lower triangular matrices are non singulee if

l 1 0 lnu F O

N invertible lowee triangular matrices have lower triangularinverses

v same as Gv for unit lower triangular

Proefof rest is easy Induction over matrix size n

n Abe E defE I D a e O a to be o

ni HSince L If IRM

xChH

D L L I e IRh L c O RTL tart 0

Li Li is low triangular rEtqu I

due to induction assumption

L c O D C O D L is howa triangularB